On Fej´er Type Inequalities via ( p , q ) -Calculus

: In this paper, we use ( p , q ) -integral to establish some Fej´er type inequalities. In particular, we generalize and correct existing results of quantum Fej´er type inequalities by using new techniques and showing some problematic parts of those results. Most of the inequalities presented in this paper are signiﬁcant extensions of results which appear in existing literatures.


Introduction
Quantum calculus or q-calculus, the modern name of the study of calculus without limits, has been studied since the early eighteenth century. The famous mathematician Leonhard Euler (1707-1783) established q-calculus and, in 1910, F. H. Jackson [1] determined the definite q-integral called the q-Jackson integral. Many applications of quantum calculus appear in mathematics, such as number theory, orthogonal polynomials, combinatorics, basic hypergeometric functions, and in physics, such as mechanics, relativity theory and quantum theory, see for instance [2][3][4][5][6][7][8][9][10] and the references therein. Furthermore, the fundamental knowledge and also the fundamental theoretical concepts of quantum calculus are covered in the book by V. Kac and P. Cheung [11].
The function f : [a, b] → R is called convex if for all x, y ∈ [a, b], α ∈ [0, 1], and f is called concave provided that − f is convex.

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Let I be the interval or real numbers and f : I ⊆ R → R be a convex function on I with constants a < b in I. The well-known inequality which is Hermite-Hadamard inequality [34] is In 2000, S. S. Dragomir et al. [35,36] proved related result to the Hermite-Hadamard inequality, as in the following. Theorem 1. Refs. [35,36] If f : I → R is a twice differentiable function where a, b ∈ I with a < b and real constants m and M with m ≤ f ≤ M, then and The Hermite-Hadamard inequality and the Hermite-Hadamard-Fejér inequalities, which are famous inequalities for convex functions, have a deep relationship to its integral mean, see [37][38][39][40][41][42][43][44][45][46][47] for more details and the references cited therein. A weighted generalization of inequality (1) was introduced by L. Fejé [48], as in the following.
In [49], N. Minculete and F. C. Mitroi introduced the inequalities which become important, as follows.
Some inequalities of Hermite-Hadamard-Fejér type for differentiable functions follow from Theorem 3.

Theorem 4.
Ref. [49] Let f : I → R be a twice differentiable function with a < b in I such that and In qcalculus, some Fejér type inequalities for differentiable functions were established by W. Yang [50]. Moreover, Fejér type inequalities for fractional integrals were established by M. Z. Sarikaya [51].
In this paper, we propose to generalize and extend some Fejér-type inequalities in q-integral and fractional integral to (p, q)-integral. In particular, we correct existing results of quantum Fejér-type inequalities by using new techniques and showing some problematic parts of those results. The results presented here would extend some of those in existing literatures.

Preliminaries
In this section, we give fundamental concepts of (p, q)-calculus used in our work. We will use I = [a, b] ⊆ R, I 0 = (a, b), and p, q are constants with 0 < q < p ≤ 1 throughout this paper. Definition 1. Refs. [22,23] Let f : I → R be a continuous function. The (p, q)-derivative of the function f at x on [a, b] is A function f is called (p, q)-differentiable on I if for each x ∈ I there exists a D p,q f (x). If if a = 0 in Definition 1, then 0 D p,q f = D p,q f , where D p,q f is Furthermore, if p = 1, then a D p,q f = a D q f , which is the q-derivative of the function f . where [n] p,q = p n − q n p − q .

Definition 2.
Refs. [22,23] Let f : I → R be a continuous function. The (p, q)-integral of the function f for x ∈ I is defined to be Furthermore, for c ∈ (a, x), the (p, q)-integral is defined to be If x a f (t) a d p,q t exists for each x ∈ I,, then we say f is (p, q)-integrable on I. Observe Definition 2 reduces to the q-integral of the function f when a = 0 and p = 1.

Main Results
In 2017, W. Yang [50] obtained some Fejér-type quantum integral inequalities. Unfortunately, there are many mistakes in the proofs. Many q-integrals are calculated incorrectly. Besides, the results of lemma and theorems are also wrong. Here, we will show the errors of Lemma 3 in [50]. Statement 1 (Lemma 3, [50]). If f : I → R is a twice q-differentiable function with a D 2 q f q-integrable on I, then It follows that f satisfies the conditions of Lemma 1. The left side of Equality (7) and (8) become respectively. The right side of Equality (7) becomes and the right side of Equality (8) becomes Since q ∈ (0, 1), Equalities (9) and (10) are not equal to 0. Therefore, Equality (7) and (8) are not correct.
Since Lemma 1 is used in the proof of Theorems 9 and 10 in [50], there are errors in those theorems. Now, we show that Theorem 9 in [50] is not correct. Statement 2 (Theorem 9, [50]). Let f : I → R be a twice q-differentiable function with a D 2 q f q-integrable on I, such that m ≤ a D 2 q f ≤ M. It follows that and It follows that f satisfies the conditions in Theorem 2 with −1 ≤ a D 2 q f ≤ 1. Then, we have and Also, As we seen, from (13) to (15) and for q ∈ (0, 1) we write .
That is, . Therefore, Inequality (11) is not correct. Inequality (12) also has the same error.
Next, we give some inequalities of Fejér type inequalities by using (p, q)-integral. If p = 1, then we give the correct results of Fejér type quantum integral inequalities.
Theorem 7. Let f : I → R be a twice (p, q)-differentiable function such that m ≤ f ≤ M. It follows that and Proof. Taking (p, q)-integral for Inequality (6) with respect to λ over [0, p] yields Using direct computation and variable changing in (18), we have Similarly, using (p, q)-integration on the first inequality of Theorem 3 with respect to λ over [0, p], we obtain Using direct computation and variable changing in (20), we obtain Inequality (16) comes from (19) and (21). Next, using (p, q)-integration on the second inequality of Theorem 3 with respect to λ over [0, p], we obtain Changing the variable, we have which implies Inequality (17). This completes the proof of theorem.
and m 4 Proof. Multiplying Inequality (6) by w λ (b, a), we get Taking (p, q)-integral for Inequality (24) with respect to λ over [0, p] yields Using directly computation and variable changing in (25), we obtain m 2 Similarly, multiplying the first inequality of Theorem 3 by w λ (b, a), and subsequently take (p, q)-integral on the obtained inequality with respect to λ over [0, p] yield From (27), we change the variable and apply the symmetry of w(x), it follows that m 2 Then, we obtain Inequality (22) from (26) and (28). Next, multiplying the second inequality of Theorem 3 by w λ (b, a), and subsequently taking (p, q)-integral on the obtained inequality with respect to λ over [0, p] yields By using the change of the variable of (29), we get m 8 which implies Inequality (23). The proof of the theorem is complete.
Lemma 2. If f : I → R is a twice (p, q)-differentiable function with a D 2 p,q f (p, q)-integrable on I, then and Proof. Using (p, q)-integration by parts yields which is Inequality (30).
Next, we prove Inequality (31). Using (p, q)-integration by parts, we obtain and Adding (32) and (33), we obtain which is Inequality (31). Thus the proof is completed.
Taking p = 1 in Lemma 2 yields the correct result of Statement 1. (34) and (35) become

Remark 3. From Example 3, the left side of Equality
respectively. The right side of Equality (34) becomes and the right side of Equality (35) becomes which shows the result appearing in Corollary 1.
Theorem 9. Let f : I → R be a twice (p, q)-differentiable function where a D 2 p,q f (p, q)-integrable on I with m ≤ a D 2 p,q f ≤ M. It follows that and Proof. Since m ≤ a D 2 p,q f ≤ M, it follows that Take (p, q)-integral for Inequality (38) with respect to x from a to pb + (1 − p)a, we obtain Applying Inequality (30) in Lemma 2 and into (39), we get which implies Inequality (36). From m ≤ a D 2 p,q f ≤ M, we have and for all x ∈ I. Taking (p, q)-integral on (40) and (41) with respect to x from a to pb + (1 − p)a, we obtain and respectively. By directly computation, we obtain and Substituting (44) into (42), we get And substituting (45) into (43), we get Adding (46) and (47), we obtain Substituting Equality (31) into (48), we get Inequality (37). This completes the proof.
Taking p = 1 in Theorem 9 yields the correct result of Statement 2.
Corollary 2. If f : I → R is a twice q-differentiable function with a D 2 q f q-integrable on I such that m ≤ a D 2 q f ≤ M, then

Remark 4.
From Example 4, f satisfies the conditions of Corollary 2 with −1 ≤ a D 2 q f ≤ 1. Then we have

and
(1) Also, As we seen, from (49) to (51) and for q ∈ (0, 1) we write For instance, choose q = 1 2 , we have That is, which shows the result described in Corollary 2.
Theorem 10. If f : I → R is a twice (p, q)-differentiable function with a D 2 p,q f (p, q)-integrable on I such that m ≤ a D 2 p,q f ≤ M, then Proof. We observe that Substituting b, f (x), and g(x) in Theorem 9 in [23] by pb p,q f (x), respectively, we obtain By Equality (31) in Lemma 2, we obtain which implies Inequality (52). This completes the proof.
Taking p = 1 in Theorem 10 yields the correct result of Theorem 10 in [50].
Corollary 3. If f : I → R is a twice q-differentiable function with a D 2 q f q-integrable on I and m ≤ a D 2 q f ≤ M, then

Remark 5.
If p = 1 and q → 1, then Theorem 10 reduces to the result obtained in [36].

Lemma 3.
Let φ, ϕ : I → R be two continuous and (p, q)-differentiable functions on I 0 . If a D p,q ϕ(x) = 0 on I 0 and m ≤ a D p,q φ(x)/ a D p,q ϕ(x) ≤ M on I 0 , then Proof. If a D p,q ϕ(x) > 0, then ϕ(x) is an increasing function with for all x ∈ I 0 . For a ≤ x ≤ y ≤ b, taking (p, q)-integral for Inequality (54) from x to y with respect to x yields

Multiplying the inequality above by
Similarly, if a D p,q ϕ(x) < 0, then we also obtain (55). Taking (p, q)-integral for Inequality (55) from a to pb − (1 − p)a with respect to x and y, we have and which is Inequality (53). This completes the proof.
Theorem 11. Let f : I → R be a twice (p, q)-differentiable function with a D 2 p,q f (p, q)-integrable on I such that m ≤ a D 2 p,q f ≤ M. It follows that Proof. Let φ(x) = a D p,q f (x) and ϕ(x) = x − a + b 2 .
which implies Inequality (59). The proof is complete.

Conclusions
We have established some inequalities of Fejér-type inequalities by using (p, q)-integral, such as the trapezoid-like inequalities, the midpoint-like inequalities, the Fejér-like inequalities. In particular, we generalized and corrected existing results of quantum Fejér-type inequalities by using new techniques and showing some problematic parts of those results. Our work improves the results of Fejér-type quantum integral inequalities. By taking q → 1 and p = 1, our results give classical inequalities. The (p, q)-integral inequalities deduced in the present research are very general and helpful in error estimations involved in various approximation processes. With these contributions, we hope that these techniques and ideas established in this article will inspire the interest of readers in exploring the field of (p, q)-integral inequalities.